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We introduce a new model of the nonlocal wave equations with a logarithmic damping mechanism. We consider the Cauchy poroblem for the new model in the whole space. We study the asymptotic profile and optimal decay and blowup rates of solutions as time goes to infinity. The damping terms considered in this paper is not studied so far, and in the low frequency parameters the damping is rather weakly effective than that of well-studied fractional type of nonlocal damping. In order to get the optimal estimates in time we meet the so-called hypergeometric functions with special parameters.

Asymptotic profile and optimal decay of solutions of some wave equations with logarithmic damping